Continuity of the bending map
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 1, pp. 93-119.

L’application de plissage d’une variété hyperbolique de dimension 3 associe à une métrique hyperbolique convexe cocompacte sur une variété compacte à bord sa lamination géodésique mesurée de plissage. Il a été démontré dans [KeS] et [KaT] que cette application est continue. Dans ce texte, on étudie l’extension de cette application à l’espace des métriques hyperboliques géométriquement finies. On introduit une relation d’équivalence dans l’espace des laminations géodésiques mesurées et on montre que l’application quotient de l’application de plissage est continue.

The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a 3-manifold with boundary to its bending measured geodesic lamination. As proved in [KeS] and [KaT], this map is continuous. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the space of measured geodesic laminations and show that the quotient map obtained from the bending map is continuous.

@article{AFST_2008_6_17_1_93_0,
     author = {Lecuire, Cyril},
     title = {Continuity of the bending map},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {93--119},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 17},
     number = {1},
     year = {2008},
     doi = {10.5802/afst.1178},
     mrnumber = {2464096},
     zbl = {1158.53027},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1178/}
}
Lecuire, Cyril. Continuity of the bending map. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 1, pp. 93-119. doi : 10.5802/afst.1178. http://www.numdam.org/articles/10.5802/afst.1178/

[AnC] Anderson (J.W.) and Canary (R.D.).— Algebraic limits of Kleinian groups which rearrange the pages of a book, Invent. Math. 126, 205-214 (1996). | MR 1411128 | Zbl 0874.57012

[BeP] Benedetti (R.) and Petronio (C.).— Lectures on hyperbolic geometry, (1992). | MR 1219310 | Zbl 0768.51018

[BiS] Birman (J. S.) and Series (C.).— Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24, no. 2, 217-225 (1985). | MR 793185 | Zbl 0568.57006

[Bo1] Bonahon (F.).— Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124, 71-158 (1986). | MR 847953 | Zbl 0671.57008

[Bo2] Bonahon (F.).— Variations of the boundary of 3-dimensionnal hyperbolic convex cores, J. Diff. Geom. 50, 1-24 (1998). | MR 1678469 | Zbl 0937.53020

[Bo3] Bonahon (F.).— Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form, Ann. Fac. Sci. Toulouse Math. 5, 233-297 (1996). | Numdam | Zbl 0880.57005

[BoO] Bonahon (F.) and Otal (J.-P.).— Laminations mesurées de plissage des variétés hyperboliques de dimension 3, Ann. Math. (2) 160, No.3, 1013-1055 (2005). | MR 2144972 | Zbl 1083.57023

[Br] Bridgeman (M.).— Average bending of convex pleated planes in hyperbolic three-space, Invent. Math. 132, 381–391 (1998). | MR 1621436 | Zbl 0912.30028

[CEG] Canary (R.D.), Epstein (D.B.A.) and Green (P.).— Notes on notes of Thurston, Analytical and Geometrical Aspects of hyperbolic Space, 3-92 (1987). | MR 903850 | Zbl 0612.57009

[EpM] Epstein (D.B.A.) and Marden (A.).— Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and Geometric Aspects of Hyperbolic Space, 113-253 (1987). | MR 903852 | Zbl 0612.57010

[Ga] Gabai (D.).— On the geometric and topological rigidity of hyperbolic 3-manifolds, J. Amer. Math. Soc. 10 , 37-74 (1997). | MR 1354958 | Zbl 0870.57014

[Jor] Jorgensen (T.).— On discrete groups of Möbius transformations, Amer. J. Math. 98, 739-749 (1976). | MR 427627 | Zbl 0336.30007

[KaT] Kamishima (Y.), Tan (S. P.).— Deformation spaces on geometric structures, Aspects of low-dimensional manifolds, Adv. Stud. Pure Math. 20, 263-299 (1992). | MR 1208313 | Zbl 0798.53030

[KeS] Keen (L.) and Series (C.).— Continuity of convex hull boundaries, Pac. J. Math. 127, 457-519 (1988). | Zbl 0838.30043

[Le1] Lecuire (C.).— Plissage des variété hyperboliques de dimension 3, Inventiones Mathematicae 164, no. 1, 85-141 (2006). | MR 2207784 | Zbl 1097.57017

[Le2] Lecuire (C.).— Bending map and strong convergence, preprint.

[Ot1] Otal (J.-P.).— Sur la dégénérescence des groupes de Schottky, Duke Math. J. 74, 777-792 (1994). | MR 1277954 | Zbl 0828.57008

[Ot2] Otal (J.-P.).— Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996). | Zbl 0855.57003

[Se] Series (C.).— Quasifuchsian groups with small bending, Warwick preprint (2002).

[Ta] Taylor (E.).— Geometric finiteness and the convergence of Kleinian groups, Com. Anal. Geom. 5, 497-533 (1997). | MR 1487726 | Zbl 0896.20033

[Th] Thurston (W.P.).— The topology and geometry of 3-manifolds, Notes de cours, Université de Princeton (1976-79).